Cosets/Additive case/Z and vector space/Example

In an (additively written) commutative group like ${}\mathbb {Z}$ or a vector space ${}V$, and of a given subgroup ${}H$, the equivalence relation ${}x\sim _{H}y$ means ${}y-x\in H$, that is, there exists a ${}h\in H$ such that

${}y=x+h\,.$

The equivalence classes are of the form ${}x+H={\left\{x+h\mid h\in H\right\}}$. In case ${}H=\mathbb {Z} d\subseteq \mathbb {Z}$ with a fixed ${}d$, the equivalence classes have the form

$H=\mathbb {Z} d,\,1+H=\{\ldots ,1-d,1,1+d,1+2d,\ldots \},\,2+H=\{\ldots ,2-d,2,2+d,2+2d,\ldots \},\ldots .$

These classes encompass those integer numbers that have, upon division by ${}d$, the remainder ${}0$, or ${}1$, or ${}2$, etc. They form a partition of ${}\mathbb {Z}$.

The equivalence classes of a linear subspace.

If ${}H=U\subseteq V$ is a linear subspace, then the equivalence classes have the form ${}v+U={\left\{v+u\mid u\in U\right\}}$ for a vector ${}v\in V$. This is ther affine space with starting point ${}v$ and the translating space ${}U$ (in the sense of definition). The equivalence classes form a family of affine subspaces parallel to each other.